• No results found

A moving mesh finite element algorithm for fluid flow problems with moving boundaries

N/A
N/A
Protected

Academic year: 2019

Share "A moving mesh finite element algorithm for fluid flow problems with moving boundaries"

Copied!
8
0
0

Loading.... (view fulltext now)

Full text

(1)

This is a repository copy of

A moving mesh finite element algorithm for fluid flow problems

with moving boundaries

.

White Rose Research Online URL for this paper:

http://eprints.whiterose.ac.uk/1762/

Article:

Baines, M.J., Hubbard, M.E. and Jimack, P.K. (2005) A moving mesh finite element

algorithm for fluid flow problems with moving boundaries. International Journal for

Numerical Methods in Fluids, 47 (10-11). pp. 1077-1083. ISSN 0271-2091

https://doi.org/10.1002/fld.860

[email protected]

https://eprints.whiterose.ac.uk/

Reuse

See Attached

Takedown

If you consider content in White Rose Research Online to be in breach of UK law, please notify us by

(2)

White Rose Consortium ePrints Repository

http://eprints.whiterose.ac.uk/

This is an author produced version of a paper published in International Journal

for Numerical Methods in Fluids.

White Rose Repository URL for this paper:

http://eprints.whiterose.ac.uk/1762/

Published paper

Baines, M.J., Hubbard, M.E. and Jimack, P.K. (2005) A moving mesh finite

element algorithm for fluid flow problems with moving boundaries. International

Journal for Numerical Methods in Fluids, 47 (10-11). pp. 1077-1083.

(3)

$&%('*)+,).- /1032546)7825':9,);</>=@?BACBDBDFE,GIHIH.JLKNM>O PQ42SR>TU42N?V/1AN=W%>XZY[U\U]^:_1`acbde5fI254&AN=:gB%ihj>kkFjclmkBnclioUprqFo)WkUoSs

t uwvxzy|{}~u€‚„ƒy|{ry6…‡†‰ˆ6Š‹{Œ…tŽˆ5}Œv3yc…‚rŠ v3‘ƒˆ|’ry5“wƒˆ|v”

•

v3–8ˆ6Š‹—”˜yc…‚wuwvxzy|{}~™—v3’r{r“8š›y5

œ˜žQŸ #¡¢Q£I¤ K,¥

œ‘ ¦§¨©ªª  .«¬ Ci­¯®

 #¢¬—°3 ± ž¡²" #³U´

C

µ3¶

2R1TB4&'·0325%('gS¸78TU'W9,260¹TB'·=:º5AS»½¼%(=WqF264&AN=:'@¾ŒgS¸¿2ÀT?B=W%>XF»½ÁV9>=W'*2c½%(=XF9>'·A»P ÃÄg5Ōjj>k5»Ä264ÀÆBAS9>=W4S2&»¿½ÇÈZÈcÉÊ» ¼.Â3)mË!Ìmºc91g6gB<gÀ¸ŒÍgU0RI/1'·=W%>XF»½¼%(=WqF254&AÀ=:'·¾!gS¸ÎÏ2N2N?BAS»ÎÏ2N2N?BAS»ÐÎ#ÌijznB+mÑÏ»›¼.Â3)

ÒIӌÔÕÔֽר

ÖÚÙrÛUÜIÝWÞmß3ÙràFá&âVã.ÞÏÝLäcàà|åWà|ÙràFÞ(äÙrà5äcâÏÛIæVݯá çmècÛ>ç#Û1á&àBæzé@Û>è äcâÏàëê1æmê>çiäcÝWÜ1à á&Û1å¯ìmäcݯÛ1ÞrÛéá&àFíFÛ>Þ.ærê>ÞÏæVé·Û>ìmè&äcâÛè6æmà|è ÙrÛUÜiݯÞmßZî#Û>ìÏÞÏæÏêè&ïçmècÛ1îmå¯àFÙráðëâÏݯí6âà5ñmâmݯîÏÝLä á&íBêåWà3ÝWÞIÜ>êècÝWê>ÞmíFà1òmóâÏà›àBô(ì.êUäcÝWÛ>ÞÏáé@Û>è äcâmà3ÙràFá&â8ÙrÛUÜ,àFÙràFÞ(äêècà î.êá&àBæõìmç#Û1ÞöäcâÏàzå¯Û(íBêåêçÏçÏå¯Ý¯íBêUäcÝWÛ>Þ÷Ûéëêá&íBêå¯à|øÝWÞIÜ>êècÝWê>Þ(ä!íFÛ>ÞÏá&à|ècÜ>êäcݯÛ>Þ÷çmècݯÞmíFݯçÏå¯àݯÞÏíFÛècç#Û>è6êUäcÝWÞmßöêÙrÛ>ÞÏÝLäcÛ>è é·ìmÞÏí|äcݯÛ1ÞùêÞ.æúâ.êBÜ,àûî#àFàFޗá&ìÏí|íFàFá&áNé·ìmå¯å¯ïùêçÏçmåWݯàBæ‰äcÛüçmècÛ>îÏå¯àFÙráýݯޗî#Û>äcâ—Û>ÞÏàõê>Þ.æ‰äSðÛ"á&ç.êíFàþæiÝWÙrà|ÞÏá&ݯÛ1ÞmáFò

ÿ

ñmêÙrçÏå¯àFá½êècà!çmècÛUÜIÝWæmàBæ§äcÛá&âÏÛBðùäcâÏàŒç#à|è&é@Û>ècÙ8ê>ÞmíFà!Û>éQäcâmàŒçmècÛ1ç#Û>á&àBæÕê>å¯ß>Û>ècÝLäcâÏهìÏá&ݯÞÏßêÙrÛ1ÞÏÝLäcÛè½é@ìÏÞmí|äcݯÛ1Þ î.êá&àBæýìÏç#Û>ÞêUècí|øåWà|ÞÏß>äcâòÛ>çIïIècݯß1â(ä í Û1âmÞúݯå¯à|ï‘ÒiÛ1Þmáä6æò

!

ÔýÛUÜIݯÞÏßî#Û>ìÏÞ.æmêècݯàFá"mÙrÛUÜiݯÞmßzÙràFá&âmàFá"mãÏÞÏÝLäcà¹àFå¯àFÙrà|ÞIäëÙrà5äcâÏÛIæ"iá&íBêå¯àZݯÞIÜ>êècÝWê>ÞmíFà

#%$'&)(*,+.-0/01023*'&4-0(

57689;:=<>@?A4B=?ABCD8E?F=GH68%?IKJH6KLM?C8DC%NPOCQL=N;?JH6R6ENS9;: 2 6E>FOJHCQJ49;6%:C%NUTUN;O=9;I /V :C%>9XW AYMBCD89P:=< LM? ?:A4OWWZ?AHA\[]O=NPNV C%F=F=N;9P?I^9P:>_CQ: V I9P`a? GH? :RJWZ6E:EJH?ZbJHAYGCQ:=<E9P:=<[]GH6%>cFBC%A4?WBCQ:<%?dCQ:I^L=N;6e'fgOF F=GH6%LNP?>@AhJ46iB V F?G4LM6%N;9;WjWZ6E:A\?G48CQJ49;6%:kNXCDe.AlC%:Im>@6%GH?K<E? :=?GHC%NnW N;CEA4A4?Al6Q[,JH9P>@?ZfoI?F?:I? :RJ@p6e $ &:qJ4B=9XAhFCQFM? G@Cr>@689P:=<s>@?A4Bkt:=9PJ4?^? N;? >@? :RJ>? J4B=6Iq9;AF=GH?A4? :RJ4?Ii[]6%GlJ4B=?A46%N;OJ49;6%:m6%[CuW N;CEA4A 6%[vAHW C%NP? fg9;:8DC%G49XCQ:RJ,FC%G\JH9;C%NwI9x`a? GH? :RJH9;C%Ny?zROCQJ49;6%:A|{~} /0 A€'e,9PJ4B7>@689P:=<@LM6%O=:I=CQGH9P?A $ NPJ4B=6EO=<%B7:=6 C%:CQNV A\9XA|9XAhF=GH?A4? :RJH?I‚YyJ4B=?^:O=>@? GH9;WCQNƒ? bF?G49;>@? :RJHAlA\O=<E<%?A)JJHBCJlJ4B=?^CQF=FG46RC%WB„?ZbB=9;L=9PJHAA\9;>@9PNXCQG A\JHC%L=9PN;9PJV F=GH6%FM? G4J49;?A3J46_A\JHC%:I=CQGI‚Yt=b?Ij>@?A\BwYt:=9xJH?? N;? >@?:EJ,>@?ZJHB=6I=A $

* B=?>689;:=<>@?A4BsC%F=F=GH6ECEWB7BC%A…L?? :sG4?†R9;:IN;?IrL V GH?W ? :RJ…9;:RJ4? GH?A\J…9;:u<E? 6E>? J4GH9;Wh9P:RJH? <%GCJH9P6E: C%:I_A4WCQN;?.9;:R8C%G49XCQ:W ?%YQe,B9;WB@JHG4?CJHAƒ9P:I? FM? :I=? :RJ3C%:I_I? FM? :I? :RJn8C%G49XCQL=N;?AƒCQN;9P†E?h‡ˆQ‰$R&:@J4B=9XAnFCQFM? G JHB=?@>?A\Bs?zROCQJ49;6%:AdCQGH?hLC%A4?Ir6E:uJHB=?@F=GH9P:W 9PFNP?6Q[nW 6%:A4? GH89P:<J4B?9;:RJ4?<%GCQNU6Q[3C^AHW CQN;?h9;:8CQGH9;C%:EJ >@6%:9xJH6%G_[]O=:WŠJ49;6%:‹9;:ŒJ49;>@?ue,9PJ4B=9;:‹?CEWBŒFCJWB‹6Q[0t:=9PJ4?u?NP?>@? :RJHA $ƒ :C%I=I9PJ49;6%:CQNW 6%:A\J4GCQ9;:RJ@9XA GH?zRO=9;G4?I_JH6A4F?WZ9P[

V

J4B?0>@?A\Bj8%? N;6WZ9PJ

V

O=:=9XzRO=? N

V

YJHB=9;A3LM? 9;:=<_W C%G4GH9P?I_6%O=J3JHB=GH6%O=<EBjCl>?A\Bj8%?NP6WZ9PJ

V

FM6QJH? :RJ49XCQN'9;:qJHB=?j>_C%:=:=? G@6%[l‡Ž‰ $1

:=NP9;†%?^>@6EA\JCQF=F=GH6ECEWB=?AJ46i>@68R9;:=<sL6EO=:I=C%G

V

F=GH6%L=N;? >_AYUJ4B=? C%F=F=GH6Db9;>_CJH9P6E:hF=GH6WZ?I=O=G4?'OA4?AJ4B=?} /… JH6|6%LJCQ9;:lJHB=?.>@?A4B@8%?NP6WZ9PJ49;?ALCEA\?IlOF6E:@C%:_CQF=F=GH6ECEWB

‘

‚’Š“~“g”)•g–Q’Š—D˜”\—™\”š~’,J›,”)œQU™)’Š›,–RžŸ ”)”)˜D•4ž¡™Šž¢D£

(4)

JHBCJƒBC%A C%NPGH?CEI V LM? ? :A4OWWZ?AHA\[]O=NPNV CQF=F=N;9;?IlJ46hCdGCQ:<%?'6Q[‚>@689P:=<dLM6%O=:I=CQG V F=GH6%LNP?>@A 9P:_6E:=?.CQ:I J)e36_A\FCEWZ?|I9;>@? :A49P6E:A'OA49P:<lJHB=?I?F?:I? :RJ.8CQGH9XCQL=N;?|C%A'>@6%:=9PJ46EG'[]O=:WZJ49;6%:wYy‡# ‰$

WCQN;9P:<,9;ASC':CJHO=GHC%NQF=GH6%FM? G4J

V

6Q[>@6I? NXA‚6%[FB

V

A49;WCQN%A

V

A)JH? >_AwIO?J46,JHB=? 9;Gw9;:I?F?:I? :WZ?ƒ6%[RFB

V

A49;WCQN O=:9xJA‡ ‰

$

T6%G.C_AHW C%NP?09P:8CQGH9XCQ:RJ,F=G46EL=N;? > JHB=? GH?d?Zb9;A\J.9;:I9XWZ?A! kC%:I#"uA\OWBjJ4BCJ,J4B?A4WCQN;9P:=<

$&%('*)$+-,.%('0/

)

,1+324%5'76

)

2

{ # €

N;?CD8E?AnJ4B=?} /…

281%:9;2 { €

{e,B=? GH? 9&2

9;AnC|F=O=GH? NV A4FCJH9;C%N6EF?GHCQJ46%G6E:CQ:_?8%6%N;89P:<dI=6%>_CQ9;:=<d{

$

€4€ CQ:I_CQF=F=GH6%FG49XCJH?3LM6%O:I=CQG V W 6%:I9PJ49;6%:Al9;:8CQGH9;C%:EJ $ T6%G?Zb=CQ>@F=N;?%Yv9;:mJHB=?7W C%A4?K6%[.J4B?^FM6%GH6%OAh>@?I=9PO=> ?zROCJH9P6E:{}n5  €h9;:?> I9;>@? :A49;6%:AYe,B=9XWB^GH? FG4?A\?:EJA39;A4? :RJ4GH6%F9;W0<ECEAnp6e JHB=G46EO=<%B^FM6%GH6%OA3>@?I9XC=Y

2 8 %(@BA

{

2CD@E2

€

+

A4O=L0F)?WŠJ,JH6

2*GHJI.%:KD+

{MLE€

9PJ3WCQ:LM?…A4B=6e,:@J4BCJ!

%

#N {POQ>SR € CQ:IT" %BU

> N {POQ>VR €ŠY%e,B9PN;?.[]6EG J4B?[]6EO=G\JHBf¤6%GI? G ?zROCJH9P6E:

2 8

R

@WA

{

2CD@X@TYZ2

€

%5K7+

A4O=L0F)?WZJ,J46

21GH[I.%]\

2

\

O

GHJI^%5K7+

{ˆR€

% #N {OQ>VRuˆR€ƒCQ:I_" %`U > N {OQ>VRuˆR€ $ GCQ:=<E?.6Q[UCQF=F=N;9XW CJH9P6E:A 6%[wJ4B9;A3?zROCQJ49;6%:KWCQ:LM?…[]6%O=:IK9P: ‡a‰w{C%:IlG4? []? GH? :W ?AyJ4B=?G4?9P:€ $ T=6EGSLM6QJHBl6%[=J4B=?A\?'F=GH6%LNP?>@ASJHB=? GH?n? b9;A\JU†:=6e,:A\?Nx[foA49P>@9;N;C%GUA46%N;OJ49;6%:A Y ‡cbYDaQ‰gY=e,B9;WBCQGH?d9;I?CQNa[]6EG.WZ6%>@FC%G49XA46%:Ke,9PJ4B^JHB=?|G4?A\ONxJA'6%LJCQ9;:=?IjL

V

:O=>@? GH9;WCQN‚AHWB=? >@?A

$

def^g7higkjl0mhng7opqsr&gDoZmhng

t 9;8%?: CQ: 9;:=9xJH9;C%N.WZ6E:I9PJ49;6%:m[]6EG7{ €ZY CsA4?ZJ_6%[J4?A)J@[]O=:WŠJ49;6%:AXu!v4YCQ:I Cs:=6%:f¤:=?<ECJH9P8E?%YvA\6ENPO=J49;6%:f I?F?:I?:EJ.>@6E:=9xJH6%G'[]O=:WŠJ49;6%:4w„{

2x+y@E2

€ŠYJHB=? :6%:?0>_C V I=?Zt:=?z v1{„‡ K7+

# ‰yA\OWBjJ4BCJ

|

I

u}vw„{ 2x+y@E2

€D>e<

%

zv

|

I

wi{ 2+@E2

€0>~< %€

v

+

A4C VE$

{gŽ%€

&[3J4B=?JH?A\J|[]O=:WŠJH9P6E:Ad[]6EG4> CjFC%G\JH9xJH9P6E:s6%[3O=:9xJV JHB=? :‚

v

z v

%

#E$ T=O=G4J4B? GH>6EG4?EYwWZ6E:A\?G48CQJ49;6%:r6%[ {gŽ%€ŠY‚CEAJHB=?_A\6ENPO=J49;6%:

2

CQ:IsI6E>_CQ9;:ƒ<d{

$

€? 8E6%N;8%?l9P:sJ49;>@?%Y‚>_C V L?@OA4?IsC%A…J4B=?<%O9;I9;:=<^F=GH9P:W 9PFNP? []6EG…C@>@?A4Br>@68%?>?:RJCQN;<%6%GH9PJ4B=> {C%A9P:m‡# ‰Ue,9xJHB.w„{

2x+y@E2 €V„

2

€ $ T6%G…C_AHW C%NP?h9P:8CQGH9;C%:RJ.F=G46EL=N;? > {gŽ%€3>_C V LM?|>6I9Pt?IKJH6@L?WZ6%>@?

|

I…8P† u}vˆ‡wi{

$+‰2x+y@E2 €D>e<

%Š

v {MaE€

e,B=?G4?0JHB=?pu}v CQGH?0AHW C%NP?d9;:8DC%G49XCQ:RJ.CQ:I‹‡w 9XA'<%9;8%?:jL V

‡

w„{$+‰2x+y@E2 € %€$ŒŽ/ w„{$Œ6 2x+‰$Œ6/0@E2 € {‘bQ€

( 6QJ4?JHBCJ:‡w

%

w CQJ $ˆ%

# {]e,B=9XWBwY‚e,9PJ4B=6EOJ|NP6RA4A06Q[n<%? :? GCQN;9xJ V Y9XA0JHC%†%? :uJ46^L?JHB=?@9P:=9PJ49XCQNvJH9P>@? JHB=G46EO=<%B6%OJ0J4B=9XAdFC%F?G€…CQ:IuJ4BCJYwCEAdCjGH?A4O=NPJ06%[3AHW C%NP?9P:8CQGH9XCQ:W ?%YJ4B=?



v C%G4?l:=6e 9;:I? FM? :I? :RJ 6%[

$

$’ ?_A4BCQN;NvG4? []? G…J46^?zROCJH9P6E:q{aR€0C%AJ4B=?_WZ6E:A4? GH8DCQJ49;6%:7F=GH9;:WZ9;F=N;? $w* B=9XA0?zROCJH9P6E:uA4O=<%<E?A\JHAJ4B=? ? b9XA\J4? :WZ?,6Q[‚Cd>_CQF=F9P:=<

,

{

$

€U[]6%Gƒe,B=9;WB_AHW C%NP?'9;:R8C%G49XCQ:W ?'9;A A4OA\JHC%9P:=?Il[]6%GƒC%NPN $V“

#E$’ ?.:=?ZbJƒI?G49;8%? JHB=?|8%?NP6WZ9PJ V•”

,

{

$

€3?ZbF=N;9;W 9xJHNV L V I9x`a? GH? :RJH9;CQJ49;:=<r{aR€ƒe,9PJ4BGH?A4F?WŠJ,JH6

$

$

a’Š–Z–“n— ¦ŠœZš ™

˜

(5)

ü8U©ÚQ>" * ûÏ þ© *

DEZgJmh ”

,

{

$

€

1 A\9;:=<w?9PL:=9AyG4ONP?.{~CQ†C3J4B=? + ? V :=6EN;I=A * GCQ:A4FM6%G4J * B=? 6EG4?>_€aC%:IdC%AHA\O>9;:=<'JHBCJ H H 8 R ” ,&Ac@ u v %5K {9$? $ JHB=?dJ4?A)J,[]O:WŠJH9P6E: u v 9XA,C%I=8%?WZJ4?IKe,9PJ4B8%?NP6W 9xJV ”

, €ZY=e3?|6%LJCQ9;:K[]GH6%> {MaE€ K % > > $ | I…8P† u v w„{‡

$+Ž2+@E2 €0>~< % | IQ…8P† > > $

{Pu vw€‡ R @`A

{Pu vw‡ ”

, €`>e< % | IQ…8P† u}v \ ‡ w \ $ R \ ‡ w \ 2 R \ ‡ w \ @E2 A[@ \

2 \ $ R @BA {7‡w ” , € >~<› {E€ & [,e3?:6e A4O=LA\J49PJ4OJH?_[]6%G \ 2 N \ $ []GH6%> J4B=?} /… { €…JHB=9;AhL?WZ6E>?A|CQ:„?zROCQJ49;6%:i[]6%G ” , $3V 9PJHA4? NP[ JHB=9;Ah9;Ah9P:A\OW 9P?:EJhJ46rI? J4? GH>@9P:?

”

,

O=:=9XzRO=? NV 9;:m>@6%GH?@J4BC%:„6%:=?KA4FCEWZ?I9;>@? :A49P6E: $ 6en?8%?GY‚L V JHB=? ? N;>@B=6%NPJ / ?W 6%>@FM6EA49xJH9P6E: * B=?6%GH? > O=:9;zRO=?:=?AHAv>_C V L?'6ELJHC%9P:=?IhL V CEI=I9PJ49;6%:C%NPNV A4F?WZ9P[V 9;:=< 2 "!

”

,

CQ:IrCKA4O=9PJHCQLNP?lL6EO=:I=C%G V WZ6E:I9PJ49;6%: $3V e,GH9PJ49;:=< 2 "!

”

, %B2 #!#$

YMe,B? GH?

$

9XA…F=G4?A4W G49;LM?I‚Y 9PJ…[]6ENPN;6e.A,JHBCJ…J4B=?G4?l?Zb9XA)JA…CKF6%J4? :RJH9;C%Ny[]O=:WŠJ49;6%:&%„A\OWB7J4BCQJ ”

, %'$

R

@

% $ { 9P:W ?le3?lA4BC%NPNS:=6QJ BCD8E?.6W WC%A49P6E:@J46hOA\?0C:=6E:f ? GH6

$

9;:Ke,BCJƒ[]6ENPN;6e.A 9PJ'9;A3A\? JƒJH6(? GH6Y%9;>@F=NV 9;:=<CQ:9;G4GH6QJCJH9P6E:CQN ”

,

$€  zROCQJ49;6%:„{)E€n>_C V :=6eL?|e,GH9PJ\J4?:7C%A,Cen?CQ†_[]6EG4> 6Q[UC%:? N;NP9;FJ49XWd?zROCQJ49;6%:j[]6%G*%SY

U

|

IQ…8P†

u

v

@BA {7‡w @ %a€0>~< %

| I…8P† u v \ ‡ w \ $ R \ ‡ w \ 2 R \ ‡ w \ @2

AJ@ 9&2 >~<› {+E€

 W 6%:8%?:=9P?:RJ3e3?C%†_[]6%GH>c6Q[UJ4B?|?zROCJH9P6E:A'WZ6E:=:=?WZJ49;:=< ” , CQ:I,%79XA | I…8P† u v {

” ,4U @ %a€.-k>~< %ŠK7+ []6%G z % # +  +

> { #

K

€

’ ?|G4? []? G'JH6{+E€'C%:Is{#

K

€'C%A3J4B?|F6%J4? :RJH9;C%NwCQ:I^8%?NP6W 9xJ V ?zEOCJ49;6%:A'G4?A\FM?WZJ49;8%? NV%$

/pj;7hiEZ10XZ7hMo

T6%N;NP6e,9;:=<@‡# ‰gYN;?ZJ ,3254 Y%C0F=9;?WZ?e,9;A4?'NP9;:=?CQGUt:=9PJ4?,?NP?>?:RJv>_C%F=F=9;:=<d[]G46E> A46%>@?nGH?Z[]?G4?:WZ?,I6E>_CQ9;: {]JV F9;WCQN;NV <d{# €4€ $3* B=9;AKI?Zt:?AC„>@689P:<st:=9xJH?r? N;? >@?:EJ>@?A\BŒ6%:Œe,B=9XWB u

v

276

v

YJ4B?rOA4OCQN F=9;?W ? e,9XA\?iN;9P:=?CQGLCEA\9XAj[]O=:WŠJH9P6E: CJ:=6I?&8HY'e,B=9;NXA)J9% 2;: CQ:I 2<2>= CQGH?sF=9;?W ? e,9XA\?iN;9P:=?CQG C%F=F=GH6Db9;>_CJH9P6E:A

$*

B=?W 6%:A4? GH8CJ49;6%:jF=GH9P:W 9PFNP?{MaE€ƒJ4B=?:7L?WZ6E>?A

|

I…8P†

6

v w„{‡ $+?=!+@@=

€0>e< %BA

v { #%# €

AHC V Ye,B=?G4?…J4B=?

A

vCQGH?0†R:6e,:[]GH6%> JHB=?d9P:=9PJ49XCQN‚>@?A\BCQ:I^I=CQJHC $

9;>@9PNXCQGHNV YJ4B?0FM6QJH? :RJ49XCQN‚?zROCJH9P6E: {+E€3>_C V LM?|?ZbF=GH?AHA\?IKC%A

| I…8P† ‡ w @@6 v A@@: >~< % | IQ…8P† 6 v \ ‡ w \ $ R \ ‡ w \ = R \ ‡ w \ @C=

AJ@ 9D= >~< + { #J € e,B=?G4? : % K

BCEA@L?? :ŒCQF=FNP9;?I 6%: J4B=?LM6%O=:I=CQG V {W 6%GHG4?A\FM6%:I=9P:=<sJ46kC9 ?G46sJHC%:=<%?:EJH9;C%N'>?A\B 8E? N;6W 9xJV CJ'J4B=?|LM6%O:I=CQG V € $=* B=?|8E? N;6WZ9PJV ?zROCQJ49;6%:„{ #

K €nL?WZ6%>@?A | I…8P† 6 v { ”

4 U @C: €.-›>~< %ŠK7+ []6%G z % # + + > + { # LE€

W 6%GHG4?A\FM6%:I=9P:=<lJ46JHB=?|L?A)JC%F=F=GH6Db9P>_CJH9P6E:

”

4

JH6 @C:

9P:^JHB=?A\FC%WZ?|A4FCQ::=?IjL V J4B=?

6

v $

a’Š–Z–“n— ¦ŠœZš ™

˜

(6)

ž(Dž Qi­,ž,yž( ür§ž3ž(B

1 A49P:<uJHB=?Kt:=9xJH?? N;? >@? :RJ@?ZbFCQ:A\9;6%:A 4 %



4 6 +?: % 

: 6 +?= % 

= 6 YUJ4B=? >_CJHG49Pb@[]6%GH>_Aƒ6%[w?zROCJH9P6E:A0{ #J € C%:Ir{# LE€ƒW CQ:KL?dI? GH9;8%?I $R* B=?A\?…?zROCQJ49;6%:Aƒ[]6%GH> J4B=?0LC%A49XAƒ6%[‚J4B=? >@?ZJHB=6Ihe,B? GH? L V YD<%9;8%?: = 6%:lC0>@?A4B 4 YQC>@?A\B8%?NP6WZ9PJV

”

4 W C%:hLM?n[]6EO=:I $Q* B9;AU9XAvOA\?I|JH60OFMI=CQJ4? JHB=?|>@?A4B^89XC[]6%GHe3C%GHI  O=N;? G'JH9P>@? f¤A\J4?F=F=9;:=<r{AHC V €ŠY=C[JH? G.e,B9;WB^J4B?A\6ENPOJH9P6E:^WCQ:LM?|G4?WZ68%?G4?I6E: JHB=?d:=? e >@?A4Bj89XCJHB=?WZ6E:A\?G48CQJ49;6%:F=GH9;:WZ9;F=N;?{#%# € $ NPJ4?G4:CJ49;8%?NV Y = >_C V C%N;A46lLM?CQFF=G46Db9;>_CJ4?I OA49;:=<J49;>? f¤A\J4?F=F=9;:=<_LCEA\?IKO=FM6%:^JHB=?|en?CQ†@[]6EG4> | IQ…8P† 6 v ” = >e< % | I…8P† 6 v { @C=5A ” 4 R 9 =

€0>~< { # ˆR€

e,9PJ4B

”

= %:K

6E:KJHB=?|L6EO=:I=C%G V 6Q[*<d{

$

€ $

$ 5 -0(.&)*-0+ T 1(d23*'&4-0(

* B=?WZ6E:A\?zRO=? :W ?A6Q[ JCQ†9P:<_w J46KLM?lJ4B=?4I?:A49xJV >@6%:9xJH6%G.[]O:WŠJH9P6E:

2

BCD8%?LM? ?:u? bRJH? :A49;8%? NV A\J4OI=9P?I 9;: ‡# ‰$ 5rCQ: V 6%J4B=?G_F6RA4A49PLNP?7WB=6E9;W ?A[]6%G w CQGH?^FM6EAHA49PL=N;?B=6e3? 8%?G $ T6%G@J4B?7G4?>@C%9P:I? G 6%[,J4B=9XAFCQFM? Gen?WZ6E:A49;I?G F)OA\J@6%:=?j6Q[.JHB=?A4?%YSJ4B=?4C%GHWZfgN;? :<QJ4B >@6%:=9PJ46EGj{]e,9XI?NV OA4?IqL?W C%OA\? 6%[09PJHA@JH? :I=? :W

V

J46k>@68E?^:6I=?A_9;: J46kGH? <%9;6%:A@e,B=? GH?^J4B=?uA46%N;OJH9P6E: <EGHCEI9P?:RJ@9XA_B=9;<%BM€<%9;8%? :ŒL

V # R { @E2 € Y

Y=CQNPJ4B6%O=<EBJ4B?0<E? :=?GHC%NP9XAHCJ49;6%:@JH6l6%J4B=?G'>6E:=9PJ46%GAn[]6%N;NP6e.An9;:^CA\9;>@9PNXCQG'>_CQ::=? G

$ T=GH6%> {‘bQ€ƒ‡w %:$ ŒŽ/ # R $ Y … /DŒ6 † { @2 € Y

C%:Ij?zROCJH9P6E:i{ #J €nJ4B=?:^LM?W 6%>@?A

| IQ…8P† $ Œ Y … /DŒ6 †

R {@C= €

Y

@@6

v

A[@C: >e< %

|

I…8P†

6

v

$

{U > #R { U "w€\€

$ Œ Y … /DŒ6 †

R {@C= €

Y >~< R | IQ… 8P† 6 v U { U "w€ $ Œ Y … /DŒ6 † Œ R @C=(AJ@ { 9 = € $ Œ Y … /7Œ6 † R { @C= € Y >e< + { # Ž%€ e,B=9;N;?{ #%# €nL?WZ6%>@?A | I…8P† 6 v $ ŒŽ/ # R $ Y … /DŒ6 † { @C= € Y >e< %>A

vސ { # aE€

L $, }n} &423*'&4-0(

eD}gygl~okf^yql0ml7hMgSf

1 A\9;:=<J4B=?|8C%NPO=?A,6Q[x kCQ:I#"r:=6%J4?Ij9;:

?WŠJH9P6E: # e,9PJ4B^JHB=?CQGWŠf¤NP?:=<QJHBK>@6%:9xJH6%GYw{ # ŽE€3<%9;8%?A

| IQ…8P† $ Œ Y …  † …

C  Y

† R { @C= € Y @@6 v A@C: >~< % | I…8P† 6 v $ Œ OQ>kR $ Œ Y …  † …

C  Y

† R { @C= € Y >~< R | I…8P† 6 v U {>ˆR # € $ Œ …P… C  Y

†

 † …

C  Y

† N {POQ>›R €QR @@=(A@ $ Œ Y …  † …

C  Y

† R{ @C= € Y >~< + { # bQ€

e,B=9;N;?{# aR€nL?WZ6%>@?A

| I…8P† 6 v $ Œ …

C  Y

† # R $ Y …  † …

C  Y

† { @@= € Y >~< %BA

v  { # E€

a’Š–Z–“n— ¦ŠœZš ™

˜

(7)

ü8U©ÚQ>" * ûÏ þ© *

( 6QJ4? J4BCQJUIO=?ƒJ46,JHB=?ƒF9P?WZ? e,9XA4? NP9;:=?CQGSC%F=F=GH6Db9P>_CJH9P6E:…9PJS9XAw:?WZ?A4AHCQG V JH6.9;:EJHG46IOWZ?nC%:|9P:RJH? GH>?I9XCJ4? t:9xJH?|? N;? >@? :RJ,[]O=:WZJ49;6%:

2:9 =

9;:k{# b%€ŠYGH?W 68%? GH?I[]G46E> J4B?|en?CQ†@[]6%GH>

|

IQ…8P†

6

v

>e< % U

|

IQ…8P† =kC}@@6

v

A[@C=

>e<› { # +E€

+

?A4O=NPJHAh[]6%GlJ4B=?j6%:=? f¤I=9P>@? :A\9;6%:C%Nƒ?zEOCJ49;6%: CQGH?KA\B6e,:k9P: TS9;<%OG4? #E$S&

: ?CEWBmW C%A4?_JHB=?jGH?A4O=NPJHA A4B=6e,:e3? GH?6ELJHC%9P:=?I_OA49P:<^{

#

ˆR€J46O=FaI=CJH?JHB=?…8C%NPO=?Aƒ6%[wJ4B?0I?F?:I?:EJ,8CQGH9XCQL=N;?CQ:I_J4B?…9;:=9PJ49XCQN >@?A4B7e'C%A,O=:9x[]6EG4>@N

V

A4FCEWZ?I $*

B=?JH?A\J…W CEA\?lA\B=6e,:>@6I? NXACKA\9;>@9PNXCQGH9xJ

V

A\6ENPO=J49;6%:J46J4B=?}ƒ5



6%[ JHB=?.[]6EG4> <%9;8%?:_9P:s‡bY # ‰$e’ B=? :=O

%

# JHB=?…AHWB=? >@?.? bB=9PL9xJA3W NP6RA\?.J46lA4?W 6%:I@6EGHI=? GnCEW W O=GHCEW V YQe,B=9;NP? e,B=?:‚O

%

{C%:IkB=9;<%B=?G€|J4B=?^?Zb=CEWŠJA46%N;OJH9P6E:mBC%Al9;:t:=9PJ4?^<%GC%I9;? :RJlCQJhJHB=?jLM6%O:I=CQG V CQ:IkJ4B=? :O=>@? GH9XW CQN‚6EGHI=? G'6Q[vC%W W O=GC%W V GH?I=OWZ?A3J46CQFF=G46Db9;>_CJ4?NV 6E:=? $

9;>9;NXCQGƒGH?A4O=NPJHAnCQGH?.A\?? :_9;:_J)en6I=9P>@? :A\9;6%:Aƒe,B=? :jWZ6%>@FC%G49XA46%:AC%G4?,>_C%I=?e,9PJ4B_?Zb=CEWŠJnA49P>@9;N;C%G49PJ V A46%N;OJ49;6%:A $ TU9;<%O=GH? I=6R?A':=6QJ.A4B=6eC@A49P>@9;N;C%G49PJV A46%N;OJ49;6%:^B=6e3? 8%?GY9;:A\J4?CEIK9PJ.BCEA'L?? :7WB=6EA4? :KJ46 9;NPN;OA\J4GCJH?J4B?l>@68%?>@? :RJ6Q[ƒJ4B=?>@?A4B7J46e'CQGI=ACKG4?<%9;6%:m{]J4B=?>@68R9;:=<LM6%O:I=CQG V 9P:rJ4B=9XAdW C%A4?€.9P: e,B=9XWB7JHB=?h<EGHCEI9P?:RJ6Q[vJ4B?l?8%6%N;89P:<A\6ENPOJH9P6E:79;A0A\J4? ?F?:=9;:=< $a* B=?lW 6%:A4? GH8CJH9P6E:^6%[ƒCQGWŠf¤NP?:=<QJHBrW C%: W NP?CQGHN

V

L?@A4? ? :rJ46^NP?C%IJH6jCGH?I=OWŠJH9P6E:r9P:uJ4B=?>@?A4BuA49?h9;:rJ4B?lGH? <E9P6E:A.e,B=?G4?hJ4B=?<%GC%I=9P?:EJ…BC%A 9;:WZGH?CEA\?IKA4BCQGHF=N

V

68%?G3J49;>?

$

−1.5

−1

−0.5

0

0.5

1

1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u

Porous Medium Equation: n=1

Approximate

Exact

−1.5

0

−1

−0.5

0

0.5

1

1.5

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u

Porous Medium Equation: n=2

Approximate

Exact

−1.5

−1

−0.5

0

0.5

1

1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

x

u

Fourth Order Diffusion: n=1

Approximate

Exact

ݯß1ìmècà1òÐÒIÞ.ê>çmá&âÏÛäcá¹Û>éÛ1Þmà|øSæmݯÙràFÞÏá&ݯÛ>Þ.ê>åÐècàFá&ìÏåLäcáŒêUäŒÜ1êUècݯÛ1ìÏá›äcÝWÙrà|á3ݯå¯åWìmáNä&è6êäcݯÞÏßÔ

ÿ

ðëÝLäcâ *å¯à|é:ä4" Ô

ÿ

ðëÝLäcâ *ÙrÝWæÏæiå¯à4" äcâýÛè6æmà|èëðëÝLäcâ ·ècÝWß>â(ä5ò

edemgl0h‰gŽq m Zl7hng

&:l6%GI? GSJ46dCQF=FNV JHB=?'F=G46EF6RA\?ICQN;<%6EG49PJ4B>JH6JHB=?3[]6%O=G4J4B=fg6EGHI?GyF=GH6%LNP?>c{]ˆR€yOA49P:<…F=9;?W ? e,9XA\?3N;9P:=?CQG t:9xJH?|? N;? >@? :RJHA.9xJ.9XA':=?W ?AHA4C%G V JH6@?ZbF=GH?AHA'9xJCEA,ClFCQ9;G,6Q[UA4?W 6%:Ij6%GI?G,?zROCJH9P6E:A

2 8

R

@`A

{

2 C @

€

%5K7+ #% @ Y 2

 {

K

€

 A'e,9xJHBKJHB=?|}ƒ5  YC%F=F=GH6%F=GH9XCJ4?…8CQN;O=?A'6%[ iC%:I="k{~A\??

?WZJ49;6%: # €ƒ>@C V LM?A4O=LA\J49PJ4OJH?Ij9P:RJ467{ # Ž%€ 9;:_6%GI? G J46|6%L=JHCQ9;:_?zROCQJ49;6%:A[]6%GJHB=?>@?A4B_FM6QJ4?:RJ49XCQN[]O=:WŠJ49;6%:

:

$E <RCQ9;:@9xJ39;A:?WZ?A4AHCQG V JH6|G4?F=NXC%WZ? 9 =

L V Ce3?CQ†jCQF=F=GH6Db9P>_CQJ49;6%:wY

Y9;:^J4B=9XA.W CEA\?d<E9P8E? :KL V

|

IQ…8P†

6

v

>e< % U

|

I…8P† =kC}@@6

v

AJ@ >e<

+

{# €

a’Š–Z–“n— ¦ŠœZš ™

˜

(8)

ž(Dž Qi­,ž,yž( ür§ž3ž(B

e,B=?G4?

9;A3JHB=?dt:=9PJ4?|?NP?>@? :RJCQF=FG46Db9;>@CQJ49;6%:KJ46

<%9;8%?:KL V

|

I…8P†

6

v

>~< % U

|

I…8P† @@6

v

AJ@C=

>~<› { €

TU9P<EO=G4? # A\B6e.A 6%:?A\? Jn6%[wG4?A\O=NPJHA []6%GƒJ4B=9XA []6EO=G4J4B6EGHI?G ?zROCQJ49;6%:KC%:I_WZ6E>@FCQGH?AvJHB=? > e,9PJ4BJ4B=? ? bCEWŠJ,A\9;>@9PNXCQGH9PJV A\6ENPOJH9P6E:K<E9P8E? :K9;:„‡aY # ‰we,B=? : O

%

#E$* B?0:O=>@?G49XW C%NMGH?A4O=NPJHA'A4O=<%<E?A\J'C%:K6%GI?Gn6%[ CEW WZOGHCEW V 6Q[UL? J)en?? : # CQ:I 9;:^6E:=?|I9;>?:A49P6E:^C%:ICQF=F=GH6Db9P>_CQJ4? NV7# 9P:^J)e36I=9P>@? :A\9;6%:A $

−0.5

0

0.5

0

0.25

0.5

0.75

1

Porous Medium Equation: n=2

x

u

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Porous Medium Equation: n=2

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Porous Medium Equation: n=2

ݯß1ìmècà òÏÒiÞÏê>çÏá&âmÛ>äcáÛéäSðÛ>øSæiÝWÙrà|ÞÏá&ݯÛ1ÞÏê>å#ècàFá&ìÏåLäcáëêUä Ü1êUècݯÛ1ìÏáäcݯÙràFáÝWå¯å¯ìÏáNä&è6êUäcݯÞÏßzäcâmà3àFÜ,Û>å¯ìmäcݯÛ1Þ§Û>éÐäcâÏà Ô

ÿ

á&Û1å¯ìmäcݯÛ>ÞýðëÝLäcâ é:ècÛ1٠ݯÞÏÝLäcÝWê>åíFÛ1ÞÏæmÝLäcݯÛ1ÞmáëäcâÏê使iÛÞÏÛäëíFÛ>è&ècàFá&ç#Û>Þ.æäcÛzêzá&ݯÙrݯå:êUècÝLäÀïá&Û1å¯ìmäcݯÛ1Þ Ïá&ìmíFíFàFá&á&ݯÜ,à á&å¯Ý¯íFàFáëäcâmècÛ1ìmß1â

*å¯à|é:ä4".ݯÞÏÝLäcÝWê>åÙràFá&â *ÙrÝWæmæmå¯à4"Iã.ÞÏê>åÙràFá&â ·ècݯß>âIä5ò

ˆ $'/…&

231 D

&4-0(

’ ?…BCD8%?FG4?A\?:EJH?ICh>@68R9;:=<l>?A\BKt:=9PJ4?0? N;? >@? :RJ'>@?ZJ4B6IjLCEA\?I_6%:_JHB=?dOA4?6Q[UChAHW CQN;?9;:8CQGH9;C%:EJ W 6%:A4? GH8CJ49;6%:hF=G49;:W 9PF=N;?39P:W 6%GHF6EGHCQJ49;:=<…C%:lC%GHWƒNP?:=<QJHBl>@6%:9xJH6%GS[]O=:WŠJH9P6E: $ V >@>@?ZJHG49XWnW 6%>@F=OJCJH9P6E:CQN GH?A4O=NPJHABCD8%?jL?? : 9P:WZN;OI?IwY A\9;>@F=NV J46i9PN;NPOA)JHGHCQJ4?jJV F9;WCQN'L?BCD89P6EO=G@CQ:I F?G\[]6EG4>_C%:WZ?K[]6%GJ4B=9XA >@?ZJHB=6I‚YLOJA4WCQN;?|9P:8CQGH9XCQ:W ?dI=6R?A':=6QJ0I? FM? :I^6E:A V >@>? J4G V ‡ ‰SC%:IjJ4B=?dJH?WB=:9;zRO=?|9XA,[C%G,>6EG4? <E? :=?GHC%NPNV C%F=F=N;9;WCQL=N;?…‡# ‰$* B=? GH?ƒ9XAy:6GH?C%A46%:|e,B V 9PJvWCQ:=:=6%JSLM?nC%F=F=N;9P?I>hOWB>@6%GH? e,9;I=? NV YJ46…>6EG4? W 6%>@F=N;9;WCJ4?IK<E? 6E>? J4GH9P?A3e,9xJHB^6%J4B=?G,>@6%:=9PJ46EGHA,C%:I^6QJ4B? G,F=GH6%L=N;? >_A3?ZbB=9;L=9PJ49;:=<AHW CQN;?d9P:8CQGH9XCQ:W ? $

Ð Q

KŠž ‚¡— —D”)•r,­(y¢D¨D¨¡“~˜r­,— ›.¡™¤£.§3ž,u›,’[— —D¦n›,”\•~œ—J—šg””)Ÿ ”)›,”)— šU¡Ÿ ¦Š’Š“n—š~œD›P’Š“‚šgœ”n¡˜D¡–šn—Z”ƒ•g’ŠŸ ¢šn— ’Š— ’šM— ›,” X˜D”)–%”)—D˜”\—Zšw–¡“šn—¡ŸQ˜—%”)“g”)—Zšn—¡Ÿ%”¢¡HšM— ’Š—D• —šgœ.›,’J— —D¦ ¨Q’Š¢D—D˜D¡“n— ”)•4ž,ÉRURI<=:2N?3- />0¹254&=:ºNTB<i78TU'W9,260¹TB'·=:º5A CBDBDFE,Gšg’'¡–D–Q”4¡“\ž

Cž ‚¡“g”)— ¨DŸ¡Hš~šv©¤žmÌmºNTB<2ë$&%(q|TB4&=@TB%IºN2&»Ìm25<¸AN=W03=W<TB4&=W'·¾!TB%I?¹$N%('*254&032N?B=@TB'*2½É AN¾B0RI'*gB'·=:º5A&)Iy§K\ªŠªBOž ž aŸ¡£Š”Vx™už.7§gBq|=W%>X!7§25A9z7§26':91g5?BA¸&gB4ë- gB%(<=W%I2NTB4½P TB4&'·=:TB<

=26425%('·=:TB<5/1TB'·=@gB%(A¤ž%§œ(ÚwœD”)•n— •6­iS——Z”)“g•M—š– ’Ðy”\¡˜— —D¦,­(CBDBD1KŠž

EDž a¢D˜D˜ Q,­n§— ¦Š¦Š’š~šûžëwœD”j¦Š”)’Š›,”)š~“n— ™=— —Zšg”)¦Š“¤¡HšM— ’Š—m’d•~™4¡Ÿ ” — —Š¡“M—¡— š@’Š“g˜— —¡“–k¡—D˜m–¡“šn—¡Ÿ.˜—%”)“g”)—Zšn—¡Ÿ ”¢¡HšM— ’Š—D•\ž#+1gB/14&%ITB<gS¸!ÍgB0RI/>'*TB'·=@gB%ITB<TB%I?ŒÉRURI<=:2N?¹7§TB':912503TU'@=@º6AëCBDBD1KBG#J ªŠª5MEUCBCž

ž‚¡’!™ ­iy¢¡—D¦!™ü­iy¢D•~•g”)Ÿ Ÿ. nž(s›,’[— —D¦3›,”)•~œ0›,”ošgœ’˜d¨D¡•g”)˜d’Š—0š~œD”ƒ¦Š”\’Š›,”oš~“n— ™™)’Š—D•~”\“ Š¡HšM— ’Š—|Ÿ¡ žmÌ,$É7 +1gB/>4&%ITU<gB%Ìmº5=:25%('·=!º!ÍÐgB0Ri/>'·=W%>XzCBDBDBC1G"$##JLKBK\«5MIKcEUCž

Ož x— ”%ëB›­v’Š—˜— ™ ­a”)“~š~’&%%—.nž©UŸ ’Ѝ¡Ÿ›,’ ˜D”)Ÿ •';’Š“w›,’[— —D¦n™\’Š—Zšg¡™)šUŸc— —D”)•4ž(P9>¾BA¿26q(‰CBDBD1KBG*)*+.JD1KBK6CBDŠ«ž ,ž d¢D“g“g¡‰–ZUnžm78TU'W9,260¹TB'·=:ºNTB<mÄ =:gU<gNXB¾h#É%V$&%('@4Sg6?B/,º6'·=@gB%.- “g˜0”)˜—šM— ’Š—*/¤ži–D“M— —D¦Š”)“0-@CBDBDBC/¤ž

a’Š–Z–“n— ¦ŠœZš ™

˜

References

Related documents

The maximum wrist extension torque was significantly higher when the forearm was in the pronation position than when it was in the supination or neutral position (P &lt; 0.01)..

As a result, much of the literature on learning in games focuses on payoff-based schemes that only require players to observe the stream of their in-game payoffs: for instance,

Colonel Brian Beers, Commander, 18th Maintenance Group, commented “The V2 CLS leadership team provides over and above support to ensure mission success: Combat Ready Aircraft

The main objective of this paper is to analyse the impact of natural resource revenue and capital expenditure on welfare indicators (income inequality and labor

Secret sharing based secure multiparty computation is based on the principle of sharing an input secret α among n players and evaluate the polynomial arithmetic function f based

Isao Kuroda, then a professor at the School of Human Sciences of Waseda University, suggested the introduc- tion of risk management technology developed in the field of aviation

The rural banking system in Ghana was, therefore, established to address the challenges associated with the provision of institutional credit and other